Question

Determine o valor de p de modo que a equação 2x²-2(p-1)x+5p=0 não admita raízes reais.

Answer 1

$\boxed{\mathrm{2x^2-2(p-1)x+5p=0}}\ \to\ \mathrm{a=2\ \ \|\ \ b=-2(p-1)\ \ \|\ \ c=5p}\\\\ \mathbf{Para\ que\ n\~ao\ haja\ raizes\ reais}\ \to\ \boxed{\mathrm{\Delta\ \textless \ 0}}\\\\ \mathrm{b^2-4ac\ \textless \ 0\ \to\ b^2\ \textless \ 4ac\ \to\ [-2(p-1)]^2\ \textless \ 4.2.5p\ \to}\\ \mathrm{\to\ 4(p^2-2p+1)\ \textless \ 40p\ \to\ p^2-2p+1\ \textless \ 10p\ \to\ p^2-12p+1\ \textless \ 0}\\\\ \mathrm{p=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-(-12)\pm\sqrt{(-12)^2-4.1.1}}{2.1}=}\\\\ \mathrm{=\dfrac{12\pm\sqrt{144-4}}{2}=\dfrac{12\pm\sqrt{2^2.35}}{2}=\dfrac{12\pm2\sqrt{35}}{2}=6\pm\sqrt{35}}$

$\mathbf{Nesta\ inequa\c{c}\~ao,\ p\ \textgreater \ menor\ raiz\ e\ p\ \textless \ maior\ raiz:}\\\\ \boxed{\mathrm{p\ \textgreater \ 6-\sqrt{35}}}\ \ \mathrm{e}\ \ \boxed{\mathrm{p\ \textless \ 6+\sqrt{35}}}\ \to\ \boxed{\boxed{\mathbf{6-\sqrt{35}\ \textless \ p\ \textless \ 6+\sqrt{35}}}}$